LECTURE 26: FEEDBACK CONTROL

1 / 17

# LECTURE 26: FEEDBACK CONTROL - PowerPoint PPT Presentation

LECTURE 26: FEEDBACK CONTROL. Objectives: Typical Feedback System Feedback Example Feedback as Compensation Proportional Feedback Applications

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## LECTURE 26: FEEDBACK CONTROL

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

LECTURE 26: FEEDBACK CONTROL

• Objectives:Typical Feedback SystemFeedback ExampleFeedback as CompensationProportional FeedbackApplications
• Resources:MIT 6.003: Lecture 20MIT 6.003: Lecture 21Wiki: Control SystemsBrit: Feedback ControlJC: Crash CourseWiki: Root LocusWiki: Inverted Pendulum CJC: Inverted Pendulum

Audio:

URL:

A Typical Feedback System

Feed Forward

Feedback

• Why use feedback?
• Reducing Nonlinearities
• Reducing Sensitivity to Uncertainties and Variability
• Stabilizing Unstable Systems
• Reducing Effects of Disturbances
• Tracking
• Shaping System Response Characteristics (bandwidth/speed)

Motivating Example

• Open loop system: aim and shoot.
• What happens if you miss?
• Can you automate the correctionprocess?
• Closed-loop system: automatically adjusts until the proper coordinates are achieved.
• Issues: speed of adjustment, inertia, momentum, stability, …

System Function For A Closed-Loop System

• The transfer function of thissystem can be derived usingprinciples we learned inChapter 6:
• Black’s Formula: Closed-loop transfer function is given by:
• Forward Gain: total gain of the forward path from the inputto the output, where the gain of a summer is 1.
• Loop Gain: total gain along the closed loop shared by all systems.

Loop

The Use Of Feedback As Compensation

• Assume the open loopgain is very large(e.g., op amp):

 Independent of P(s)

• The closed-loop gain depends only on the passive components (R1 and R2) and is independent of the open-loop gain of the op amp.

Stabilization of an Unstable System

• If P(s) is unstable, can westabilize the system byinserting controllers?
• Design C(s) and G(s) so thatthe poles of Q(s) are in the LHP:
• Example: Proportional Feedback (C(s) = K)
• The overall system gain is:
• The transfer function is stable for K > 2.
• Hence, we can adjust K until the system is stable.

Second-Order Unstable System

• Try proportional feedback:
• One of the poles is at
• Unstable for all values of K.
• Try damping, a term proportional to :
• This system is stable as long as:
• K2 > 0: sufficient damping force
• K1 > 4: sufficient gain
• Using damping and feedback, we have stabilized a second-order unstable system.

The Concept of a Root Locus

• Recall our simple control systemwith transfer function:
• The controllers C(s) and G(s) can bedesigned to stabilize the system, but that could involve a multidimensional optimization. Instead, we would like a simpler, more intuitive approach to understand the behavior of this system.
• Recall the stability of the system depends on the poles of 1 + C(s)G(s)P(s).
• A root locus, in its most general form, is simply a plot of how the poles of our transfer function vary as the parameters of C(s) and G(s) are varied.
• The classic root locus problem involves a simplified system:

Closed-loop poles are the same.

Example: First-Order System

• Consider a simple first-order system:
• The pole is at s0 = -(2+K). Vary Kfrom 0 to :
• Observation: improper adjustment of the gain can cause the overall system to become unstable.

Becomes less stable

Becomes more stable

Example: Second-Order System With Proportional Control

• Using Black’s Formula:
• How does the step responsevary as a function of the gain, K?
• Note that as K increases, thesystem goes from too little gainto too much gain.

How Do The Poles Move?

Desired Response

• Can we generalize this analysis to systems of arbitrary complexity?
• Fortunately, MATLAB has support for generation of the root locus:
• num = [1];
• den = [1 101 101]; (assuming K = 1)
• P = tf(num, den);
• rlocus(P);

Summary

• Introduced the concept of system control using feedback.
• Demonstrated how we can stabilize first-order systems using simple proportional feedback, and second-order systems using damping (derivative proportional feedback).
• Why did we not simply cancel the poles?
• In real systems we never know the exact locations of the poles. Slight errors in predicting these values can be fatal.
• Disturbances between the two systems can cause instability.
• There are many ways we can use feedback to control systems including feedback that adapts over time to changes in the system or environment.
• Discussed an application of feedback control involving stabilization of an inverted pendulum.

More General Case

• Assume no pole/zero cancellation in G(s)H(s):
• Closed-loop poles are the roots of:
• It is much easier to plot the root locus for high-order polynomials because we can usually determine critical points of the plot from limiting cases(e.g., K= 0, ), and then connect the critical points using some simple rules.
• The root locus is defined as traces of s for unity gain:
• Some general rules:
• At K= 0, G(s0)H(s0) =   s0are the poles of G(s)H(s).
• At K= , G(s0)H(s0) = 0 s0are the zeroes of G(s)H(s).
• Rule #1: start at a pole at K= 0 and end at a zero at K= .
• Rule #2: (K  0) number of zeroes and poles to the right of the locus point must be odd.

Inverted Pendulum

• Pendulum which has its mass above its pivot point.
• It is often implemented with the pivot point mounted on a cart that can move horizontally.
• A normal pendulum is stable when hanging downwards, an inverted pendulum is inherently unstable.
• Must be actively balanced in order to remain upright, either by applying a torque at the pivot point or by moving the pivot point horizontally (Wiki).

Feedback System – Use Proportional Derivative Control

• Equations describing the physics:
• The poles of the system are inherentlyunstable.
• Feedback control can be used to stabilize both the angle and position.
• Other approaches involve oscillatingthe support up and down.